Graph norms and Sidorenko’s conjecture

Let H and G be two finite graphs. Define hH(G) to be the number of homomorphisms from H to G. The function hH(·) extends in a natural way to a function from the set of symmetric matrices to ℝ such that for AG, the adjacency matrix of a graph G, we have hH(AG) = hH(G). Let m be the number of edges of H. It is easy to see that when H is the cycle of length 2n, then hH(·)1/m is the 2n-th Schatten-von Neumann norm. We investigate a question of Lovász that asks for a characterization of graphs H for which the function hH(·)1/m is a norm.We prove that hH(·)1/m is a norm if and only if a Hölder type inequality holds for H. We use this inequality to prove both positive and negative results, showing that hH(·)1/m is a norm for certain classes of graphs, and giving some necessary conditions on the structure of H when hH(·)1/m is a norm. As an application we use the inequality to verify a conjecture of Sidorenko for certain graphs including hypercubes. In fact, for such graphs we can prove statements that are much stronger than the assertion of Sidorenko’s conjecture.We also investigate the hH(·)1/m norms from a Banach space theoretic point of view, determining their moduli of smoothness and convexity. This generalizes the previously known result for the 2n-th Schatten-von Neumann norms.

[1]  John von Neumann,et al.  The Cross-Space of Linear Transformations. III , 1946 .

[2]  Quantum Chemistry. Methods and Applications. , 1961 .

[3]  G. R. Blakley,et al.  A Hölder type inequality for symmetric matrices with nonnegative entries , 1965 .

[4]  E. Szemerédi.  On sets of integers containing no four elements in arithmetic progression , 1969 .

[5]  Joram Lindenstrauss,et al.  Classical Banach spaces , 1973 .

[6]  N. Tomczak-Jaegermann.  The moduli of smoothness and convexity and the Rademacher averages of the trace classes $S_{p}$ (1≤p<∞) , 1974 .

[7]  E. Szemerédi.  On sets of integers containing k elements in arithmetic progression , 1975 .

[8]  H. Furstenberg,et al.  An ergodic Szemerédi theorem for commuting transformations , 1978 .

[9]  Miklós Simonovits,et al.  Compactness results in extremal graph theory , 1982, Comb..

[10]  Dan Amir.  Moduli of Convexity and Smoothness , 1986 .

[11]  Fan Chung Graham,et al.  Quasi-random graphs , 1988, Comb..

[12]  A. Sidorenko,et al.  Inequalities for functionals generated by bipartite graphs , 1991 .

[13]  Alexander Sidorenko,et al.  A correlation inequality for bipartite graphs , 1993, Graphs Comb..

[14]  J. Diestel,et al.  Absolutely Summing Operators , 1995 .

[15]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[16]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[17]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[18]  Vojtech Rödl,et al.  Extremal problems on set systems , 2002, Random Struct. Algorithms.

[19]  V. Rödl,et al.  Extremal problems on set systems , 2002 .

[20]  L. Lovasz,et al.  Reflection positivity, rank connectivity, and homomorphism of graphs , 2004, math/0404468.

[21]  T. Tao,et al.  The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[22]  Vojtech Rödl,et al.  Regularity Lemma for k‐uniform hypergraphs , 2004, Random Struct. Algorithms.

[23]  Parallelepipeds, nilpotent groups and Gowers norms , 2006, math/0606004.

[24]  The ergodic and combinatorial approaches to Szemerédi's theorem , 2006, math/0604456.

[25]  László Lovász,et al.  Limits of dense graph sequences , 2004, J. Comb. Theory B.

[26]  Vojtech Rödl,et al.  The counting lemma for regular k‐uniform hypergraphs , 2006, Random Struct. Algorithms.

[27]  Vojtech Rödl,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006, Random Struct. Algorithms.

[28]  Jozef Skokan,et al.  Applications of the regularity lemma for uniform hypergraphs , 2006 .

[29]  V. Sós,et al.  Counting Graph Homomorphisms , 2006 .

[30]  W. T. Gowers,et al.  Hypergraph regularity and the multidimensional Szemerédi theorem , 2007, 0710.3032.

[31]  R. A. R. A Z B O R O V.  On the minimal density of triangles in graphs , 2008 .

[32]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[33]  Ben Green,et al.  New bounds for Szemerédi's theorem, I: progressions of length 4 in finite field geometries , 2009 .

[34]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS U4-NORM , 2005, Glasgow Mathematical Journal.

[35]  Vojtech Rodi.  Some Developments in Ramsey Theory , 2010 .